Is zero vector always present in any n-dimensional space?

[Asad Raza]

In the book, Introduction to Linear Algebra, Gilbert Strang says that every time we see a space of vectors, the zero vector will be included in it.

I reckon that this is only the case if the plane passes through the origin. Else wise, how can a space contain a zero vector if it does not pass through the origin?

 

[andrewkirk]

A plane in 3D space is not a vector space unless it passes through the origin. It is an Affine Space, which is a generalisation of the concept of a vector space.

The reason that a plane that doesn't pass through the origin is not a vector space is that it is not closed under addition or scalar multiplication. Consider the plane $z=2$, which contains the vectors (0,0,2) and (1,1,2). The sum of those is (1,1,4) which is not in that plane.

 

[Asad Raza]

Can you kindly explain that how (1,1,2) lies in the plane z=2. The plane z=2 should only contain (0,0,2). No?

 

[andrewkirk]

Can you kindly explain that how (1,1,2) lies in the plane z=2. The plane z=2 should only contain (0,0,2). No?

No. The plane z=2 is the set of all points (x,y,2) where x and y are any real numbers.

 

[weirdoguy]

The plane z=2 should only contain (0,0,2). No?

If it would contain only one point it wouldn't be a plane...

 

[Mark44]

Can you kindly explain that how (1,1,2) lies in the plane z=2. The plane z=2 should only contain (0,0,2). No?

To expand on what andrewkirk said, the equation z = 2 means that both x and y are completely arbitrary. You are assuming that since x and y aren't mentioned, both must be zero. Instead, since they aren't present, they can take on any values.

 

[Lineisy Kosenkova]

I reckon that this is only the case if the plane passes through the origin. Else wise, how can a space contain a zero vector if it does not pass through the origin?

 

[Skins]

If the plane does not pass through the origin then it is not a vector space (according to definition a vector space must contain the zero vector).